A scenario for symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities

TL;DR

This paper investigates the phenomenon of symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities through numerical solutions of Euler-Lagrange equations, revealing a new mechanism beyond classical instability explanations.

Contribution

It introduces a novel scenario for symmetry breaking in these inequalities by analyzing solution branches, supported by numerical computations using Freefem++.

Findings

Identified a new symmetry breaking mechanism.

Computed minimal solution branches numerically.

Reparametrization reveals the scenario for symmetry breaking.

Abstract

The purpose of this paper is to explain the phenomenon of symmetry breaking for optimal functions in functional inequalities by the numerical computations of some well chosen solutions of the corresponding Euler-Lagrange equations. For many of those inequalities it was believed that the only source of symmetry breaking would be the instability of the symmetric optimizer in the class of all admissible functions. But recently, it was shown by an indirect argument that for some Caffarelli-Kohn-Nirenberg inequalities this conjecture was not true. In order to understand this new symmetry breaking mechanism we have computed the branch of minimal solutions for a simple problem. A reparametrization of this branch allows us to build a scenario for the new phenomenon of symmetry breaking. The computations have been performed using Freefem++.